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Controlled transport of fluid particles by microrotors in a Stokes flow using linear transfer operators

Jake Buzhardt, Phanindra Tallapragada

TL;DR

The paper addresses steering Lagrangian transport of a passive density in Stokes flow by using two fixed microrotors to generate a controllable velocity field. It presents a data-driven framework based on finite-dimensional Perron–Frobenius/Liouville operators to model density evolution and extends it to control-affine dynamics, solved with differential dynamic programming. Finite-time coherent-set analysis links the optimized transport to coherent flow structures, revealing transport barriers that guide the density toward a target while controlling dispersion. The method is demonstrated across unbounded, plane-wall, and circular-confined domains and extended to multiple densities, highlighting boundary effects and potential microfluidic applications such as targeted cargo transport and cell manipulation.

Abstract

The manipulation of a collection of fluid particles in a low Reynolds number environment has several important applications. As we demonstrate in this paper, this manipulation problem is related to the scientific question of how fluid flow structures direct Lagrangian transport. We investigate this problem of directing the transport by manipulating the flow, specifically in the Stokes flow context, by controlling the strengths of two rotors fixed in space. We demonstrate a novel dynamical systems approach for this problem and apply this method to several scenarios of Stokes flow in unbounded and bounded domains. Further, we show that the time-varying flow field produced by the optimal control can be understood in terms of dynamical structures such as coherent sets that define Lagrangian transport. We model the time evolution of the fluid particle density using finite dimensional approximations of the Liouville operators for the micro-rotor flow fields. Using these operators, the particle transport problem is framed as an optimal control problem, which we solve numerically. This framework is then applied to the problem of transporting a blob of fluid particles in domains with different boundary conditions: free space, near to a plane wall, in a circular confinement, and the transport of two distributions of particles to a common target. These examples demonstrate the effectiveness of the proposed framework and also shed light on the effects of boundaries on the ability to achieve a desired fluid transport using a rotor-driven flow.

Controlled transport of fluid particles by microrotors in a Stokes flow using linear transfer operators

TL;DR

The paper addresses steering Lagrangian transport of a passive density in Stokes flow by using two fixed microrotors to generate a controllable velocity field. It presents a data-driven framework based on finite-dimensional Perron–Frobenius/Liouville operators to model density evolution and extends it to control-affine dynamics, solved with differential dynamic programming. Finite-time coherent-set analysis links the optimized transport to coherent flow structures, revealing transport barriers that guide the density toward a target while controlling dispersion. The method is demonstrated across unbounded, plane-wall, and circular-confined domains and extended to multiple densities, highlighting boundary effects and potential microfluidic applications such as targeted cargo transport and cell manipulation.

Abstract

The manipulation of a collection of fluid particles in a low Reynolds number environment has several important applications. As we demonstrate in this paper, this manipulation problem is related to the scientific question of how fluid flow structures direct Lagrangian transport. We investigate this problem of directing the transport by manipulating the flow, specifically in the Stokes flow context, by controlling the strengths of two rotors fixed in space. We demonstrate a novel dynamical systems approach for this problem and apply this method to several scenarios of Stokes flow in unbounded and bounded domains. Further, we show that the time-varying flow field produced by the optimal control can be understood in terms of dynamical structures such as coherent sets that define Lagrangian transport. We model the time evolution of the fluid particle density using finite dimensional approximations of the Liouville operators for the micro-rotor flow fields. Using these operators, the particle transport problem is framed as an optimal control problem, which we solve numerically. This framework is then applied to the problem of transporting a blob of fluid particles in domains with different boundary conditions: free space, near to a plane wall, in a circular confinement, and the transport of two distributions of particles to a common target. These examples demonstrate the effectiveness of the proposed framework and also shed light on the effects of boundaries on the ability to achieve a desired fluid transport using a rotor-driven flow.
Paper Structure (14 sections, 1 theorem, 42 equations, 13 figures)

This paper contains 14 sections, 1 theorem, 42 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Density Transport
  3. Perron Frobenius operator and generator
  4. Numerical approximation
  5. Extension to controlled systems
  6. Propagation of moments
  7. Finite-time coherent set detection
  8. Control formulation
  9. Transport by rotors in free space
  10. Finite time coherent sets
  11. Transport by rotors near an infinite plane wall
  12. Transport by rotors within a circular boundary
  13. Transport of two densities
  14. Conclusion

Key Result

Lemma 1

Suppose the Liouville operator associated with a vector field $f_1:\mathbf{X} \mapsto \mathbb{R}^n$ is denoted by $\mathcal{L}_1$ and the Liouville operator associated with the vector field $f_2:\mathbf{X}\mapsto \mathbb{R}^n$ by $\mathcal{L}_2$, then the Liouville operator associated with the vecto

Figures (13)

  • Figure 1: Controlled transport of a distribution of fluid particles by two micro-rotors fixed at $(-1,0)$ and $(1,0)$ from an initial density $\rho(x(0)) = \mathcal{N}((1,1),0.025I_2)$ to a target mean at $(-1,-1)$ (green circle). White filled circle indicates the sample mean and black filled circle indicates the mean predicted by the proposed method. Streamlines depict the flow field produced by the rotor control at the instant shown.
  • Figure 2: Rotor controls and fluid particle distribution moments for transport by rotors in free space as shown in Fig. \ref{['fig:freespace_stream']}. Left, Top: Rotor strengths for the left rotor ($\gamma_L$) and right rotor ($\gamma_R$). Left, Bottom: Mean of particle distribution from sample shown in Fig. \ref{['fig:freespace_stream']} ('True') and as predicted using the Liouville operator ('Predicted'). Right: Second raw moment. Markers correspond to instants shown in Fig. \ref{['fig:freespace_stream']}.
  • Figure 3: Finite-time coherent sets produced by the optimal rotor-driven flow field corresponding to the case shown in Figs. \ref{['fig:freespace_stream']} and \ref{['fig:freespace_umu']}. The sequence shows the time evolution of $10^4$ particles initially placed on a uniform grid on $[-2,2]^2$ and colored according to the partition function $f_X$, as approximated by the left singular vector of $A=\frac{1}{m}\Psi_X\Psi_Y^T$. The black line in the first image depicts the $f_X=0$ level set, and the sequence shows the evolution of this line as the dataset is advected by the flow. The purple marker and lines show the mean value and level sets of the density function corresponding to values of the initial density at one and two standard deviations from the mean, respectively.
  • Figure 4: Finite-time coherent sets produced by the optimal rotor-driven flow field with initial density $\rho(x(0)) = \mathcal{N}((-0.5,1),0.025I_2)$ . The subfigures show the $10^4$ data points at the initial time $t=0$ and final time $t=5$, colored according to the 3rd left singular vector of $A$, along with the $f_X = 0$ contour (black line). Purple contours depict the mean value and level sets of the density function corresponding to values of the initial density at one and two standard deviations from the mean, respectively.
  • Figure 5: Controlled transport of a distribution of fluid particles near an infinite plane wall at $y = -1.25$ (black line) by two micro-rotors fixed at $(-1,0)$ and $(1,0)$ from an initial density $\rho(x(0)) = \mathcal{N}((1,1),0.025I_2)$ to a target mean at $(-1,-1)$ (green circle). White filled circle indicates the sample mean and black filled circle indicates the mean predicted by the proposed method. Streamlines depict the flow field produced by the rotor control at the instant shown.
  • ...and 8 more figures

Theorems & Definitions (1)

  • Lemma 1